I noticed some people, or people in some Shuls, only wear a Yarmulka in Shul. The Shul has a box of Yarmulkas when you walk in. Is this some kind of minhag of people who don’t wear Yarmulkas all day long or are these people not very observant?

Even a frum shul normally has a box of yarmulkes by the door for “guests” (typically of a baal simcha) who may either be Reform or non-frum) who either don’t have or who have forgotten a “kipa”. I rarely see it in smaller shtieblach since the guests of the baal simchas are heimish and will have a hat or kipa. I rarely see a regular at the shul criticize or shame someone who walks in w/o a yarmulke.

Although the OP is referring to Modern Orthodox shuls that have Yarmulka boxes for the non-observant Ashkenazim davening in MO shuls, I think there are still some observant Sephardim who don’t wear a Yarmulka throughout the day, but do for davening.

Joseph: I agree but with a caveat. Sephardim generally wear a kippah for davening and eating (from what I recall they are strict to wear a kippah when making brachot).

GH: I grew up in a Young Israel shul where many members were not observant and we needed to keep both kippot and talleisim in the hallway of the shul. As you stated, I never recall anybody being criticized for not having their own kippah or tallis.

To answer the OP, I stated they are likely less observant then those who always come to shul but out of respect (as T22T commented) will wear it for religious functions inside of a shul. The OP use of the word “minhag” is inappropriate for this situation.

The shul I grew up in had the custom that hats were not worn in davening, only yarmulkes. Some members who wore caps all day at work would arrive in shul, remove their caps, don a yarmulke from the box and daven. After davening they’d reverse the process.

ctl: I took the question from the OP as the men were totally bareheaded; no hats or caps. I also remember a story about a lawyer who came into shul and started davening without a hat or cap. His fellow mispallelim were startled. He couldn’t figure out what was wrong until he realized that he was wearing his toupee, which he wore because they did not allow him to wear a kippah in the office.

Wearing a yarmulke in public (kal-ve-homer a tallis katan, beard and pe’os, or for a woman a modest outfit and a hair covering) exposes one to significant discrimination (and in some places, though not the US, exposes one to potentially lethal violence). Many people who are somewhat observant (and sometimes, actually frum) are afraid to express their Jewishness in public.

At one time in the not too distant past, many non-observant Jews (in the U.S.) identified with Orthodoxy; and when they attended shul, they attended an Orthodox shul. I’m not sure that today many non-observant Jews still identify with Orthodoxy.

When I grew up, the boys generally wore various kinds of caps (like the ones all the goyim wore in the movies from the 1930s and 40s) outside of yeshiva. Today, its either yamulkes or baseball hats. Back then I recall there was considerably greater concern about “standing out” whereas b’h today’s yeshiva and day school kids (not black hatters) wear their yarmulkes with no concern